I used to be able to search for tests that were run using the public api- I added a query string param so that I could search on URL. However lately these tests aren't showing up for me, I can only access the results if I keep track of the test id.

If you log into the UI you should be able to see or search for your own tests (private or not). They should also be visible on the same browser if cookies are enabled.

Being able to find arbitrary tests depends if the tests were run as private or public tests. The default used to be for all tests to be public unless the user manually picked for it to be private. A few months ago the default was flipped so it defaults to private unless you manually select public.